## 6. The length of life of an instrument produced by a machine has a normal ditribution with a mean of 12 months and standard deviation of 2 m

Question

6. The length of life of an instrument produced by a machine has a normal ditribution with a mean of 12 months and standard deviation of 2 months. Find the probability that an instrument produced by this machine will last a) less than 7 months. b) between 7 and 12 months.

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2 weeks 2021-10-08T04:10:11+00:00 2 Answers 0

Step-by-step explanation:

Since the length of life of an instrument produced by a machine has a normal distribution, we would apply the formula for normal distribution which is expressed as

z = (x – µ)/σ

Where

x = length of life of instruments in months.

µ = mean time

σ = standard deviation

From the information given,

µ = 12 months

σ = 2 months

a) We want to find the probability that an instrument produced by this machine will last for less than 7 months. It is expressed as

P(x < 7)

For x = 7,

z = (7 – 12)/2 = – 2.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.0062

b) between 7 and 12 months is expressed as P(7 ≤ x ≤ 12)

For x = 7, the probability is 0.0062

For x = 12,

z = (12 – 12)/2 = 0

Looking at the normal distribution table, the probability corresponding to the z score is 0.5

Therefore,

P(7 ≤ x ≤ 12) = 0.5 – 0.0062 = 0.4938

(a) .

(b) .

Step-by-step explanation:

We have been given that the length of life of an instrument produced by a machine has a normal distribution with a mean of 12 months and standard deviation of 2 months.

(a) First of all, we will find z-score corresponding to sample score of 7 months as: , where,

z = Z-score,

x = Sample score, = Mean, = Standard deviation.

Upon substituting our given values in z-score formula, we will get: Now, we need to find the probability that a z-score is less than .

Using normal distribution table, we will get: Therefore, the probability that an instrument produced by this machine will last less than 7 months is .

(b) Let us find z-score corresponding to sample score of 12 months. Using formula , we will get:   Therefore, the probability that an instrument produced by this machine will last between 7 and 12 months is .